The propagation of spherical blast

The attenuation of spherical shocks at large distances from the origin is investigated mathematically. To do this a general theory is developed of spherical wave motion at large distances, where the linearized approximations of the theory of sound are valueless owing to the divergence of the characteristics at infinity. In the case of disturbances small from the outset, this general theory is used to modify the linearized approximation to give a theory which is uniformly valid at all distances from the origin. The equation of the shock is a$_{0}$t = r - b log$^{\frac{1}{2}}$ r - b$_{1}$ - b$_{2}$ log$^{-\frac{1}{2}}$ r + O (log$^{-1}$ r), where r is the distance from the origin at time t, measured in multiples of some characteristic length, a$_{0}$ is the velocity of sound in the undisturbed air and b, b$_{1}$, b$_{2}$ are arbitrary constants, of which b is readily calculated on the assumption of small disturbances. If a second shock is produced, the pressure at a point between the shocks falls approximately linearly with time at a rate {2$\gamma $/($\gamma $ + 1)} a$_{0}$r$^{-1}$ log$^{-1}$r atm./sec. For a weak explosion, the modified linear theory is used to show that behind the front shock an envelope of characteristics is formed and hence a second shock must appear.